The Chern numbers of the topological bulk gaps can be evaluated directly using existing numerical techniques developed for aperiodic systems.[1]} However, due to the large number of gaps in Fig. REF  and to the large number of (strong and weak) topological invariants per gap, an alternative high-throughput method is needed. In fact, the method based on the K-theoretic gap labels[2]}, [3]} explained below enables us to visualize the complete set of invariants associated with a gap. Besides the topological invariants, this method supplies additional predictions that can be tested against the numerical simulations and, as we shall see, this confirms beyond any doubt that the algebra of observables is indeed the non-commutative 4-torus.
